if b < -1.3462214f+10

1. Started with
   \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   20.7
2. Using strategy rm
   20.7
3. Applied frac-2neg to get
   \[\color{red}{\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{-2 \cdot a}}\]
   20.7
4. Applied simplify to get
   \[\frac{\color{red}{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{-2 \cdot a} \leadsto \frac{\color{blue}{\left(-\sqrt{{b}^2 - a \cdot \left(4 \cdot c\right)}\right) + b}}{-2 \cdot a}\]
   20.7
5. Applied taylor to get
   \[\frac{\left(-\sqrt{{b}^2 - a \cdot \left(4 \cdot c\right)}\right) + b}{-2 \cdot a} \leadsto \frac{\left(-\left(2 \cdot \frac{c \cdot a}{b} - b\right)\right) + b}{-2 \cdot a}\]
   5.8
6. Taylor expanded around -inf to get
   \[\frac{\left(-\color{red}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}\right) + b}{-2 \cdot a} \leadsto \frac{\left(-\color{blue}{\left(2 \cdot \frac{c \cdot a}{b} - b\right)}\right) + b}{-2 \cdot a}\]
   5.8
7. Applied simplify to get
   \[\frac{\left(-\left(2 \cdot \frac{c \cdot a}{b} - b\right)\right) + b}{-2 \cdot a} \leadsto \frac{b + \left(-\left(\frac{c \cdot 2}{\frac{b}{a}} - b\right)\right)}{a \cdot \left(-2\right)}\]
   1.6

---

8. Applied final simplification

if -1.3462214f+10 < b < -1.4384158f-33

1. Started with
   \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   4.0
2. Using strategy rm
   4.0
3. Applied frac-2neg to get
   \[\color{red}{\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}} \leadsto \color{blue}{\frac{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}{-2 \cdot a}}\]
   4.0
4. Applied simplify to get
   \[\frac{\color{red}{-\left(\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}{-2 \cdot a} \leadsto \frac{\color{blue}{\left(-\sqrt{{b}^2 - a \cdot \left(4 \cdot c\right)}\right) + b}}{-2 \cdot a}\]
   4.1

if -1.4384158f-33 < b < 1.6720318f+16

1. Started with
   \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   13.9
2. Using strategy rm
   13.9
3. Applied flip-+ to get
   \[\frac{\color{red}{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   15.3
4. Applied simplify to get
   \[\frac{\frac{\color{red}{{\left(-b\right)}^2 - {\left(\sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}^2}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \leadsto \frac{\frac{\color{blue}{c \cdot \left(4 \cdot a\right)}}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]
   6.5
5. Using strategy rm
   6.5
6. Applied *-un-lft-identity to get
   \[\frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{red}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a} \leadsto \frac{\frac{c \cdot \left(4 \cdot a\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]
   6.5
7. Applied times-frac to get
   \[\frac{\color{red}{\frac{c \cdot \left(4 \cdot a\right)}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a} \leadsto \frac{\color{blue}{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}}{2 \cdot a}\]
   4.0
8. Applied times-frac to get
   \[\color{red}{\frac{\frac{c}{1} \cdot \frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \leadsto \color{blue}{\frac{\frac{c}{1}}{2} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}}\]
   4.0
9. Applied simplify to get
   \[\color{red}{\frac{\frac{c}{1}}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a} \leadsto \color{blue}{\frac{c}{2}} \cdot \frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}\]
   4.0
10. Applied simplify to get
    \[\frac{c}{2} \cdot \color{red}{\frac{\frac{4 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}{a}} \leadsto \frac{c}{2} \cdot \color{blue}{\frac{4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\]
    4.0
11. Applied simplify to get
    \[\frac{c}{2} \cdot \frac{4}{\color{red}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}} \leadsto \frac{c}{2} \cdot \frac{4}{\color{blue}{\left(-b\right) - \sqrt{{b}^2 - \left(c \cdot a\right) \cdot 4}}}\]
    3.9

if 1.6720318f+16 < b

1. Started with
   \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
   29.6
2. Applied taylor to get
   \[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \leadsto \frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}\]
   7.1
3. Taylor expanded around inf to get
   \[\frac{\color{red}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a} \leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]
   7.1
4. Applied simplify to get
   \[\color{red}{\frac{-2 \cdot \frac{c \cdot a}{b}}{2 \cdot a}} \leadsto \color{blue}{\frac{c}{b} \cdot \frac{-2}{2}}\]
   0